On the days before the New Year: there is a thorough cleaning of the house: this is believed to sweep away all the bad luck from the past year. Also, on the new year, it is considered very unlucky to sweep/mop, for it is like throwing away the good luck.

On the New Year Eve, there is a big dinner, traditionally celebrated with lots of family. A dish of fish and/or chicken is also predominant. Sometimes the fish is not completely eaten and left until the next day. This is due to the phrase "年年有餘" ("May there be surpluses every year") which sounds like "May there be fish every year". If you compare this to a western Christmas dinner, the fish is like the big roasted turkey/chicken/ham. Also, something that is popular is "年糕", which also sounds like "a more prosperous year" [ Mand. NianGao, Cant, LianGo] It is also popular in the Philippines because of its large Chinese population, where it is known as tikoy. Can be white of brown, and is a sticky and glutinous food. Also, "Buddha's Delight" is commonly served for vegetarians (and Buddhists practicing the exercise of "no meat" for the first day +. ("羅漢齋").

On New Year's day, most people tend to not eat meat (especially Buddhists) as it is to ensure longevity. (haha, I never do this ^^')
All the food eaten is cooked the day before (so cook a lot on the New Year's Eve dinner so that there is lots of leftovers :] ) This is because it is considered bad luck to light fires and use knives on this day.
On this day, it is important to visit the seniors/elderly in the family. (unless you live in the west while they still live in the east x'])
Lion dance troupes are thought to evict bad luck, so many people may invite/go watch them perform.
Firecrackers are traditionally used for fun, but have been banned in some places for being a fire hazard. (in Canada, there are only two days where it is legal to light fireworks.. Canada day and Victoria Day.. aha; )
Red pockets are also given by married couples to young children and teenagers. If one is not married, they are not entitled (or supposed to) give red pockets. (Technically, even a 30-40 year old + that is not married can still receive red pockets; but they are usually not given them because it is impolite [seems like you are pointing out the fact they are not married yet])

On the second day: It is traditional for married daughters to visit their parents (as traditionally, the daughter goes to live with her husband and do not see her parents much after she is married)

Third and fourth days: Not traditional or lucky to do any visiting.  This is thought be either because of "赤口", meaning that is is easy to get into arguments due to the fried food eaten and visiting done during the first two days, or because families who have immediate kin deceased in the past three years do not go visiting as respect for the dead. (but they may be visited on this day).  This third day is given to visit the graves of the deceased instead.

Fifth day: In northern China, people eat "餃子" (dumplings) [Mand. Jiao Zi, Cant. Gao Zhi] on this morning. Also is the birthday of the Chinese god of wealth. Businesses traditionally re-open on this day in Taiwan.

Seventh day: Known as "人日" [Mand. Ren Ri, Cant. Yan Yat] (lit. Man's birthday). This is the common (everyday) man's birthday, a day in which everyone grows one year older.
For Chinese Buddhists, this is another meat-free day.
For Chinese in Malaysia and Singapore (^^), it is a day when a raw fish salad is eaten. People get together to make the salad and make wishes for continuing wealth and prosperity.

Ninth day: A day to offer prayers to the Jade Emperor of Heaven (天公) for Taoists.
The birthday of the Jade Emperor is traditionally on this day.
This day is special for Hokkiens and Teochews (MinNan speakers). At Midnight on the Eighth day, Hokkiens will offer thanksgiving prayers to the Emperor of Heaven. Offerings include sugarcane which had protected the Hokkiens from extermination hundreds of years ago. Tea is also served, as it is a traditional way to pay respect to an honoured person.

Fifteenth day: This marks the end of the Chinese New Year festivities. It is celebrated as "元宵节". A rice dumpling dessert, "湯圓" [Mand. TangYuan, Cant. TongYoon], which is a sweet glutinous ball, often stuffed with a black sesame mixture, is also eaten on this day. Candles are lit outside of houses to guide lost spirits home. This day is celebrated much like the Lantern Festival, and families walk the streets with lighted lanterns. And finally, for all the happy kids, they are allowed to open their red pockets on this day. (and count their money like little misers XD I know I do~!)